The effective permeability (μeff) was measured and calculated for composites consisting of micron- or submicron-sized nanocrystalline iron particles embedded in a nonmagnetic matrix. The intrinsic permeability of iron particles was obtained from the calculation for a random spatial distribution of magnetic domains and its analytical model is derived from the Landau-Lifshitz-Gilbert equation. In the calculation, each grain is assumed to be a single magnetic domain because of its nano size. The effective permeability was calculated using three methods—Bruggeman’s effective medium theory, extended Bruggeman’s effective medium theory with the consideration of the skin effect, and a simulation method which was developed in the present work. The skin effect was considered in our simulation work. Our simulation agrees well with the experimental data. Our work has shown clearly that the magnetic domain structure with a random spatial distribution of magnetic easy axes and the skin effect need to be considered to calculate the complex permeability of polycrystalline magnetic materials.
The Landau–Lifshitz–Gilbert (LL-G) formula has been widely used in modeling of complex permeability of magnetic thin films and particle composites. In this paper, a approach which is based on Landau–Lifshitz ferromagnetic resonance (LL-FMR) model has been developed to calculate the complex permeability of magnetic particles/insulator matrix composite. It has found that the LL-FMR model can be used to compute the intrinsic permeability of a magnetic particle with different alignments of its magnetic domains with the respect to the incident wave. Through integration with respect to its resonant frequency, this method offers a neat closed-form formula for the case of isotropic composite. Its results compared relatively well with our previous method which computes the isotropic permeability by taking the average of the permeability of a large number of randomly oriented magnetic domains with respect to incident wave based on the LL-G model. For carbonyl bcc-Fe and Fe3O4, the calculated permeability agrees relatively well with the measurement results.
Shape anisotropy is crucial for the microwave permeability of magnetic nanoparticles. In this work, we conduct a systematic study on the shape-dependent microwave permeability of magnetic nanoparticles. Three kinds of Fe3O4 nanoparticles with different geometric shapes (rod, disc, and octahedron) are fabricated by chemical methods and their permeability is investigated through both theoretical calculation and experiment. The results suggest that the rod could exhibit the highest resonance frequency (fr), while the lowest resonance frequency could be found in the octahedron. An opposite trend is observed in the initial permeability (μ0). In order to confirm the experimental permeability, we establish an improved LL-G model, which could take both the shape anisotropy and magnetic domain structure into account by introducing a local effective anisotropy field (Heff). The good agreement between the calculated and experimental permeability proves that the model can reproduce shape-dependent microwave permeability of magnetic nanoparticles. More importantly, no fitting process is involved in this model, enabling us to predict the permeability of magnetic nanostructures independent of experiment results.
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